A lot. I'm going to describe a few of them, though, to help give you some idea:
1. The lunge whip. The lunge whip is very similar to a ringmaster's whip, but with a longer "stock" or cane. It's used during lunging and groundwork as a signaling device. The lash can be very long.
2. Driving whip. A driving whip has a short stock and a long lash and is used as a reinforcement device when driving.
3. Dressage whip. A dressage whip is a whip with a long cane and a short lash. It's used by dressage riders when training advanced movements, to allow a "touch" to be made closer to a horse's hindquarters.
4. Crop. A crop is a short whip that does not have a lash. It does have a "flapper" on the end. The crop is the standard whip carried by English riders. Modern crops are made of fairly flexible plastic wrapped in either leather or synthetic material.
5. Show Cane. Show canes are carried by English riders in the show ring, primarily in Britain. They may be natural or leather covered, and are essentially a length of wood. This kind of whip is never used on the horse. Ever. It purely exists to "finish" a turnout and look good.
6. English hunting whip. An English hunting whip is also not for use on the horse. It consists of a cane similar to a show cane but with a 5 to 7 foot lash on it. The hunting whip is used to dissuade hounds from getting too close to a horse's hooves (a light flick with a lash is better for the dog than getting picked).
7. Quirt. A quirt is a rope or leather braid with a loop at one end. It's often carried by cowboys, and is left hanging on the saddle horn unless needed.
8. Over-under. A longer quirt, often used by barrel racers, but I've also seen them carried and used by muleskinners. An over-under makes a lot of noise and fuss but barely touches the horse - it's a good "incentive" for very lazy animals.
9. Jumping bat. A jumping bat is a very short crop that's normally only seen in the stadium or show jumping arena. It's used on the horse's shoulder gently to remind them to pick their hooves up.
10. Racing whip. The whip a jockey carries is a bit shorter than a crop with a much longer flapper, but is otherwise similar.
So, ten different kinds of "persuader" as we sometimes call them.